Finite Group Databasebeta

← #11754063

Subgroups — #11754063

Order 960.

496 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#92020C212yesmaximal, complemented
2#91801C211yesmaximal, characteristic, complemented
3C5×#912C211yesmaximal, characteristic, complemented
4C3×#4058C211yesmaximal, characteristic, complemented
5#91488C212yesmaximal, complemented
6#91486C212yesmaximal, complemented
7#91484C212yesmaximal, complemented
8#91265C211yesmaximal, characteristic, complemented
9#91264C211yesmaximal, characteristic, complemented
10D3×Dic20C211yesmaximal, characteristic, complemented
11D5×Dic12C211yesmaximal, characteristic, complemented
12#66105—31nomaximal, Hall
13#6819C2212yes—
14C5×#237C2212yes—
15C3×#711C2212yes—
16C2×#1204—22no—
17#6782—22no—
18#6780—22no—
19#6779—22no—
20#6771C2211yescharacteristic
21D15⋊Q8C2212yes—
22D3×Dic10C2212yes—
23D5×Dic6C2212yes—
24C3×C5×D8C2211yescharacteristic
25#6716C2212yes—
26Dic60C2211yescharacteristic
27C8×D15C2211yescharacteristic
28C5×#214C2212yes—
29C5×Dic12C2211yescharacteristic
30C5×D3×C8C2211yescharacteristic
31C3×#687C2212yes—
32C3×Dic20C2211yescharacteristic
33C3×C8×D5C2211yescharacteristic
34#6663C2212yes—
35#6662C2212yes—
36#6661C2212yes—
37#6651C2211yescharacteristic
38#6650C2211yescharacteristic
39#6649C2211yescharacteristic
40#5771—51nomaximal, Hall
41#4150—32no—
42C5×#135—31no—
43#4097—32no—
44D5×Q16—31no—
45D5×Q16—31no—
46#4061—32no—
47#4061—32no—
48#4058—31no—
49#4058D311yescharacteristic, complemented
50C2×Dic20—31no—
51#4047—31no—
52C3×C5×D4C2312yes—
53#1220—22no—
54C2×Dic15—22no—
55C4×D15C2311yescharacteristic
56Dic30C2312yes—
57C5×#67—22no—
58C2×C5×Dic3—22no—
59C4×C5×D3C2311yescharacteristic
60C5×Dic6C2312yes—
61C3×C22⋊D5—22no—
62C2×C3×Dic5—22no—
63C3×C4×D5C2311yescharacteristic
64C3×Dic10C2312yes—
65#1204—22no—
66#1204—22no—
67#1204—22no—
68#1204D411yescharacteristic
69#1203D411yescharacteristic
70#1202D411yescharacteristic
71#1201D411yescharacteristic
72#1200—22no—
73D3×Dic5—22no—
74D5×Dic3—22no—
75C3×C5×C8C2311yescharacteristic
76C3⋊#168C2311yescharacteristic
77C3×#168C2311yescharacteristic
78C5×C3⋊C8C2311yescharacteristic
79#1010—52no—
80C3×#135—51no—
81#951—52no—
82D3×Q16—51no—
83D3×Q16—51no—
84#915—52no—
85#915—52no—
86#912—51no—
87#912D511yescharacteristic, complemented
88C2×Dic12—51no—
89#901—51no—
90C5×#41—32no—
91Q8×D5—32no—
92Q8×D5—32no—
93#711—32no—
94#711—62no—
95#711—62no—
96#711C2×D312yes—
97#709—31no—
98#709—62no—
99C2×Dic10—32no—
100C2×Dic10—62no—
101C5×Q16—31no—
102C5×SD16—32no—
103C5×D8C2×D311yescharacteristic
104C2×C5×C8—31no—
105#689—32no—
106#689—32no—
107#687—32no—
108#687C2×D312yes—
109#681—31no—
110Dic20—32no—
111Dic20—31no—
112Dic20C2×D311yescharacteristic
113#676—31no—
114#676—31no—
115C8×D5—31no—
116C8×D5C2×D311yescharacteristic
117#578—151noSylow
118C22×C3×C5—22no—
119C2×D15C2×D411yescharacteristic
120C2×C5×D3C2×D411yescharacteristic
121C2×C3×D5C2×D411yescharacteristic
122C3×C4×C5C2411yescharacteristic
123Dic15—22no—
124Dic15C2×D411yescharacteristic
125C3×Dic5—22no—
126C3×Dic5C2×D411yescharacteristic
127C5×Dic3—22no—
128C5×Dic3C2×D411yescharacteristic
129C3×#41—52no—
130D3×Q8—52no—
131D3×Q8—52no—
132#237—52no—
133#237—102no—
134#237—102no—
135#237C2×D512yes—
136#235—51no—
137#235—102no—
138C2×Dic6—52no—
139C2×Dic6—102no—
140C3×Q16—51no—
141C3×SD16—52no—
142C3×D8C2×D511yescharacteristic
143C2×C3×C8—51no—
144#216—52no—
145#216—52no—
146#214—52no—
147#214C2×D512yes—
148#208—51no—
149Dic12—52no—
150Dic12—51no—
151Dic12C2×D511yescharacteristic
152ΓL(1,𝔽25)—51no—
153ΓL(1,𝔽25)—51no—
154D3×C8—51no—
155D3×C8C2×D511yescharacteristic
156C5×Q8—32no—
157C5×D4—62no—
158C5×D4C22×D312yes—
159C2×C4×C5—31no—
160C2×C4×C5—62no—
161C22⋊D5—22no—
162C22⋊D5—31no—
163C22⋊D5—31no—
164C22⋊D5—62no—
165C2×Dic5—22no—
166C2×Dic5—62no—
167C2×Dic5—62no—
168D20—31no—
169C4×D5—31no—
170C4×D5—62no—
171C4×D5—62no—
172C4×D5C22×D311yescharacteristic
173Dic10—31no—
174Dic10—32no—
175Dic10—62no—
176Dic10—62no—
177Dic10—62no—
178Dic10C22×D312yes—
179C5×C8—31no—
180C5×C8C22×D311yescharacteristic
181#168—31no—
182#168C22×D311yescharacteristic
1832−1+4—152no—
184#137—152no—
185#137—152no—
186#137—152no—
187#135—31no—
188#135—51no—
189#135—151no—
190C2×Q16—151no—
191C2×Q16—151no—
192C2×Q16—151no—
193#131—151no—
194C2×C3×C5—42no—
195C2×C3×C5C22×D411yescharacteristic
196D15—21no—
197C3×D5—21no—
198C5×D3—21no—
199C3×Q8—52no—
200C3×D4—102no—
201C3×D4C22×D512yes—
202C2×C3×C4—51no—
203C2×C3×C4—102no—
204#67—22no—
205#67—51no—
206#67—51no—
207#67—102no—
208C2×Dic3—22no—
209C2×Dic3—102no—
210C2×Dic3—102no—
211D12—51no—
212C4×D3—51no—
213C4×D3—102no—
214C4×D3—102no—
215C4×D3C22×D511yescharacteristic
216Dic6—51no—
217Dic6—52no—
218Dic6—102no—
219Dic6—102no—
220Dic6—102no—
221Dic6C22×D512yes—
222C3×C8—51no—
223C3×C8C22×D511yescharacteristic
224C3⋊C8—51no—
225C3⋊C8C22×D511yescharacteristic
226C22×C5—22no—
227C22×C5—31no—
228C2×D5—31no—
229C2×D5D3×D411yescharacteristic
230C4×C5—31no—
231C4×C5—62no—
232C4×C5C23×D311yescharacteristic
233Dic5—22no—
234Dic5—31no—
235Dic5—62no—
236Dic5D3×D411yescharacteristic
237#41—32no—
238#41—52no—
239#41—151no—
240#41—152no—
241#41—302no—
242#41—302no—
243#41—302no—
244C2×Q8—152no—
245C2×Q8—152no—
246C2×Q8—152no—
247C2×Q8—302no—
248Q16—31no—
249Q16—51no—
250Q16—152no—
251Q16—152no—
252Q16—152no—
253Q16—151no—
254SD16—32no—
255SD16—52no—
256SD16—152no—
257D8D3×D511yescharacteristic
258M16—151no—
259M16—151no—
260M16—151no—
261C2×C8—31no—
262C2×C8—51no—
263C2×C8—151no—
264C3×C5#57811yescharacteristic, complemented, Hall
265C22×C3—22no—
266C22×C3—51no—
267C2×D3—51no—
268C2×D3D4×D511yescharacteristic
269C3×C4—51no—
270C3×C4—102no—
271C3×C4C23×D511yescharacteristic
272Dic3—22no—
273Dic3—51no—
274Dic3—102no—
275Dic3D4×D511yescharacteristic
276C2×C5—42no—
277C2×C5—61no—
278C2×C5C2×D3×D411yescharacteristic
279D5—21no—
280D5—61no—
281Q8—32no—
282Q8—52no—
283Q8—151no—
284Q8—152no—
285Q8—302no—
286Q8—302no—
287Q8—302no—
288D4—62no—
289D4—102no—
290D4—151no—
291D4—151no—
292D4—151no—
293D4—302no—
294D4C2×D3×D512yes—
295C2×C4—31no—
296C2×C4—51no—
297C2×C4—62no—
298C2×C4—102no—
299C2×C4—151no—
300C2×C4—302no—
301C2×C4—302no—
302C2×C4—302no—
303C2×C4—302no—
304C8—31no—
305C8—51no—
306C8—151no—
307C8C2×D3×D511yescharacteristic
308C2×C3—42no—
309C2×C3—101no—
310C2×C3C2×D4×D511yescharacteristic
311D3—21no—
312D3—101no—
313C5#577111yescharacteristic, complemented, Sylow
314C22—22no—
315C22—31no—
316C22—51no—
317C22—151no—
318C4—31no—
319C4—51no—
320C4—62no—
321C4—102no—
322C4—151no—
323C4—302no—
324C4C22×D3×D511yescharacteristic
325C3#6610511yescharacteristic, complemented, Sylow
326C2—42no—
327C2—61no—
328C2—101no—
329C2—301no—
330C2D3×D4×D511yescharacteristic
subgroupG/H
centerC2D3×D4×D5
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC3×C5×D8C22
socleC2×C3×C5C22×D4
hypercenterD8D3×D5
hypocenterC3×C5#578
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #92020C2semidirect
2. #91801C2semidirect
3. C5×#912C2semidirect
4. C3×#4058C2semidirect
5. #91488C2semidirect
6. #91486C2semidirect
7. #91484C2semidirect
8. #91265C2semidirect
9. #91264C2semidirect
10. D3×Dic20C2semidirect
11. D5×Dic12C2semidirect
49. #4058D3semidirect
87. #912D5semidirect
264. C3×C5#578semidirect
313. C5#5771semidirect
325. C3#66105semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#920202
#918012
C5×#9122
C3×#40582
#914882
#914862
#914842
#912652
#912642
D3×Dic202
D5×Dic122
#661053
#57715